# Renormalization Group Running in the Symmetric and the Broken Symmetry Phases of the $R_{\xi } $ and the $\overline{R_{\xi }}$ Gauges

## TL;DR

This paper studies how the effective potential and pole mass evolve under the renormalization group in different gauges, revealing gauge-independent results and invariance of pole masses at one-loop order.

## Contribution

It demonstrates that the effective potential in the broken symmetry phase obeys the same RG equation across gauges when expressed via the VEV, and shows pole masses are RG invariant and gauge-independent at one-loop.

## Key findings

- Effective potential satisfies identical RG equations in different gauges when VEV is expressed as a function of parameters.
- Pole masses in the broken phase are RG invariant and gauge-independent at one-loop order.
- The results unify the understanding of RG running across different gauge choices.

## Abstract

We investigate the renormalization group (RG) running of the effective potential and the pole mass in the broken symmetry phase of the $R_{\xi }$ and the $\overline{R_{\xi }}$ gauges which have different RG running for the effective potential in the symmetric phase and show that if the vacuum expectation value (VEV) is expressed as a function of the other parameters of the theory by solving the minimization condition, then the effective potential in the broken symmetry phase in both gauges satisfies the same RG equation as one in the symmetric phase of the $\overline{R_{\xi }}$ gauge. The pole masses in the broken symmetry phase of both gauges are RG invariant with respect to the RG funtions of the symmetric phase and are shown to be the same at one-loop order.